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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.R.56a
Chapter 12, Problem 12.R.56a

53–57. Conic sections
a. Determine whether the following equations describe a parabola, an ellipse, or a hyperbola.
x²/4 + y²/25 = 1

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1
Identify the general form of the conic section equation. The given equation is \( \frac{x^2}{4} + \frac{y^2}{25} = 1 \). This resembles the standard form of conic sections centered at the origin.
Recall the standard forms: - Ellipse: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) - Hyperbola: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( -\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) - Parabola: Usually involves only one squared term, e.g., \( y^2 = 4ax \) or \( x^2 = 4ay \).
Compare the given equation to these forms. Since both \( x^2 \) and \( y^2 \) terms are positive and added together, and the equation equals 1, it matches the form of an ellipse.
Note the denominators: \(4\) and \(25\) represent \(a^2\) and \(b^2\) respectively, which are positive constants, confirming the ellipse shape.
Conclude that the equation \( \frac{x^2}{4} + \frac{y^2}{25} = 1 \) describes an ellipse.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. The main types are parabolas, ellipses, and hyperbolas, each defined by specific algebraic equations and geometric properties.
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Parabolas as Conic Sections

Standard Form of Conic Equations

Each conic section has a standard equation form: ellipses and hyperbolas involve sums or differences of squared terms set equal to 1, while parabolas have a single squared term. Recognizing these forms helps identify the conic type.
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Circles in Standard Form

Ellipse Identification Criteria

An ellipse equation typically has the form (x²/a²) + (y²/b²) = 1 with both denominators positive and the sum of squared terms. This distinguishes it from hyperbolas, which have a subtraction, and parabolas, which have only one squared term.
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Related Practice
Textbook Question

General equations for a circle Prove that the equations  

X = a cos t + b sin t,  y = c cos t + d sin t  

where a, b, c, and d are real numbers, describe a circle of radius R provided a² +c² =b² +d² = R² and ab+cd=0. 

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Textbook Question

51–52. {Use of Tech} Arc length of polar curves Find the approximate length of the following curves.

The limaçon r=3−6cosθ

Textbook Question

53–57. Conic sections

c. Find the eccentricity of the curve.

x²/4 + y²/25 = 1

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Textbook Question

3–6. Eliminating the parameter Eliminate the parameter to find a description of the following curves in terms of x and y. Give a geometric description and the positive orientation of the curve.

x = sin t - 3, y = cos t + 6; 0 ≤ t ≤ π

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Textbook Question

27–32. Polar curves Graph the following equations.


r = 3 sin 4θ

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. The polar coordinates (3, -3π/4) and (-3, π/4) describe the same point in the plane.